Mathematical modeling of acoustic and electromagnetic wave propagation and its efficient and accurate numerical simulation is a key technology for numerous engineering applications as, e.g., in detection (nondestructive testing, radar), communication (optoelectronic and wireless) and medicine (sonic imaging, tomography). An adequate model problem for the development of efficient numerical methods for such types of physical applications is the three-dimensional wave equation in unbounded exterior domains. Although this approach goes back to the early 1960s, the development of fast numerical methods for integral equations in the field of hyperbolic problems is still in its infancies compared to the multitude of fast methods for elliptic boundary integral equations. In the literature there exist essentially two approaches for the time discretization of hyperbolic integral equations - one is Lubich's convolution quadrature method and the other one is due to Bamberger/Ha-Duong, where a discontinuous Galerkin method is directly applied for the time discretization. Lubich's approach enjoys some nice stability properties and allows to apply fast methods for the time- and memory-consuming parts of the method. Therefore our research focus is in the development, analysis, and implementation of fast methods which are based on convolution quadrature in time and the Galerkin boundary element method in space.
B. N. Khoromskij, S. Sauter, A. Veit
Fast Quadrature Techniques for Retarded Potentials Based on TT/QTT Tensor Approximation
Computational Methods in Applied Mathematics, Vol 11(3), pp. 342 - 362 (2011) link, PDF
S. Sauter, A. Veit
Exact Solutions of Retarded Boundary Integral Equations
to be published in Numerische Mathematik Implementation and Examples fileadmin/math/preprints/03_11_revised.pdf